https://r2.erweima.ai/mathbot/image/1725489560616fg9vccpo.jpg

The function given in the image defines \( f: \mathbb{R} \to \mathbb{R} \), with a piecewise definition for different intervals of \( x \). The function \( f(x) \) is defined as follows:

\[
f(x) = 
\begin{cases} 
x, & -1 \leq x \leq 1 \\
- \frac{1}{x}, & x < -1 \text{ or } x > 1
\end{cases}
\]

The question seems to be asking for the domain of the function \( f \), which is the set of all values \( x \in \mathbb{R} \) for which the function is defined.

### Analysis:

1. **For \( -1 \leq x \leq 1 \)**: The function is simply \( f(x) = x \), which is defined for all values within this interval.
   
2. **For \( x < -1 \) or \( x > 1 \)**: The function becomes \( f(x) = - \frac{1}{x} \). This is also defined for all real numbers, **except** at \( x = 0 \), where division by zero would occur.

Thus, the function is defined for all real numbers except \( x = 0 \), as \( f(x) = - \frac{1}{x} \) would be undefined at this point.

### Conclusion:

The domain of \( f \) is all real numbers except \( x = 0 \), i.e., \( \mathbb{R} \setminus \{0\} \).

This corresponds to option **D**: \( (-\infty, 0) \cup (0, \infty) \).

Would you like further clarification on this, or do you have any other questions?

### Related Questions:
1. How would the graph of this piecewise function look?
2. Can we discuss the continuity of this function?
3. What is the range of the given function \( f(x) \)?
4. How would the function behave as \( x \to \infty \) and \( x \to -\infty \)?
5. Could you describe the asymptotes of this function?

### Tip:
When dealing with piecewise functions, always carefully consider each condition and whether any points cause the function to be undefined, such as division by zero.

Find the domain of the given piecewise function.

Explore how to find the domain of a piecewise function involving algebraic expressions defined over different intervals. Learn about key concepts like domain determination and piecewise analysis, suitable for Grades 11-12.

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